What are necessary and optional steps to get crystal clear aquarium water, i.e. water with as few large and micro particles and maximal visibility in the distance as possible?
Good mechanical filtration is the primary method. This can be accomplished using a number of methods. In freshwater, a canister or hang on back filter running filter floss or another media that (obviously) mechanically filters particles from the water is one of the best ways. For a saltwater fish only or a reef tank, a filter sock or sump or overflow utilizing floss or filter pads is typically the most common way to deal with it. This will take care of particulate matter to an extremely small size if the correct filter media is used. Some socks and filter floss can filter down to a few microns making the water extremely clear. I personally do not like using under-gravel filters as the other poster suggested as they add too much depth to the substrate level and have a number of risks that can literally destroy an entire tank if the water flow stops, but they utilize the entire substrate as a filter and can provide very good mechanical and biological filtration.
If the water is stained yellow or brown which is common and natural in both saltwater and freshwater planted tanks, some form of carbon is usually the best way to clear up the water color. A good quality carbon like ROX, which is an extremely high quality carbon by a specific manufacturer, works best, but in any case make sure to carefully follow the carbon usage instructions and cease using it if signs of HLLE (head and lateral line erosion) or other carbon related illness or stress occurs to fish, coral, or other inhabitants. If you're dealing with driftwood leaching tannins, or other features in the water that is causing it to stain, it's probably better not to add carbon and let it naturally leach the tannins over time and remove with water changes. There isn't an extremely effective method of speeding this process up in a tank, so might as well save the money on carbon and let it subside naturally while being removed with routine water changes.
Besides the right choice of location of the aquarium (not in the sunlight) which reduces the risk of algae growth and filtration, decorations and food in combination with the desired residents, the most promising step (after one year of trying) seems to be the installation of an undergravel filter with an Eheim Universal pump (300 l/h for a 60 l aqaurium was a good choice).
I didn't manage to figure out a way to let the undergravel be responsible for the aquarium alone, since
- it takes a lot of time to be sufficiently effective (you definitely need another running filter for the first weeks)
- 300 l/h is the smallest Eheim pump and it's not a large filter volume per time for a 60 l aquarium. If the filter is slightly blocked it starts to suck air which is not a problem, but indicates that 600 l/h is too much (whereas an Aquaclear 20 already has 470/h and is bearly sufficient for a 100 l tank). A second column can be added to the undergravel filter and be connected with a Y-piece to the pump, of course.
So, now I have the undergravel filter and a hang-on filter and since 3 weeks I don't have any chance to get any dirt in my vacuum cleaner - it's just completely clean which is the main reason for the water being cristal clear as well.
I wouldn't take the risk to run an undergravel filter only since it might block/reduce the flow rate and then it might be difficult to react fast enough and get another filter running.
 There seems to be no other pump which has two tube connections in the world. I searched for several days. You need a tube connection on the entry side because you need to pump the water out of the column of the undergravel filter net. Of course, 50 € for a simple filter with 300 l/h is five times more than compared to others, but this is about the water clarity.